A finite difference method for singularly perturbed differential-difference equations arising from a model of neuronal variability

In this paper a finite difference method is presented for singularly perturbed differential-difference equations with small shifts of mixed type (i.e., terms containing both negative shift and positive shift). Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. To handle the negative and positive shift terms, we construct a special type of mesh, so that the terms containing shift lie on nodal points after discretization. The proposed finite difference method works nicely when the shift parameters are smaller or bigger to perturbation parameter. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameters on the boundary layer behavior or oscillatory behavior of the solution of the problem.